Question

The dual of the converse of the inverse of the logical statement \(p\rightarrow (q\rightarrow r)\) is equivalent to...

• A. \(\sim [p∨(r\rightarrow q)]\)
• B. \(p∨(r\rightarrow q)\)
• C. \(\sim [p∨(q\rightarrow r)]\)
• D. \(p∨(q\rightarrow r)\)

Hint:

Do inverse, converse, then dual and rewrite implication as OR.

Answer 1

The Correct Option is A

Step 1: Chain of operations:
Inverse, then converse, then dual.

Step 2: Truth-table check:
Take $p=F,\ q=F,\ r=T$. The dual $\sim p\wedge\sim q\wedge r$ is $T\wedge T\wedge T=T$. Option (A): $r\rightarrow q=T\rightarrow F=F$, then $p\vee F=F$, then $\sim F=T$. Both agree.

Step 3: Another test:
Take $p=T,\ q=F,\ r=T$. Dual $=F\wedge\ldots=F$. Option (A): $r\rightarrow q=F$, $p\vee F=T$, $\sim T=F$. Agree.

Step 4: Option (C) test:
With $p=F,q=F,r=T$: $q\rightarrow r=T$, $F\vee T=T$, $\sim T=F$, which differs from $T$. So (C) fails.

Final Answer:
Option (A) matches in the test cases where (C) does not. \[ \boxed{A} \]